On the sum of $z$-ideals in subrings of $C(X)$

Abstract

Let $X$ be a Tychonoff space and $A\left(X\right)$ be an intermediate subalgebra of $C\left(X\right)$, i.e., $C^{\ast }\left(X\right)\subseteq A\left(X\right)\subseteq C\left(X\right)$. We show that such subrings are precisely absolutely convex subalgebras of $C\left(X\right)$. An ideal $I$ in $A\left(X\right)$ is said to be a $z_A$-ideal if $Z\left(f\right)\subseteq Z\left(g\right)$, $f\in I$ and $g\in A\left(X\right)$ imply that $g\in I$. We observe that the coincidence of $z_A$-ideals and $z$-ideals of $A\left(X\right)$ is equivalent to the equality $A\left(X\right)=C\left(X\right)$. This shows that every $z$-ideal in $A\left(X\right)$ need not be a $z_A$-ideal and this is a point which is not considered by D. Rudd in Theorem 4.1 of Michigan Math. J. 17 (1970), 139–141, or by G. Mason in Theorem 3.3 and Proposition 3.5 of Canad. Math. Bull. 23:4 (1980), 437-443. We rectify the induced misconceptions by showing that the sum of $z$-ideals in $A\left(X\right)$ is indeed a $z$-ideal in $A\left(X\right)$. Next, by studying the sum of $z$-ideals in subrings of the form $I+ℝ$ of $C\left(X\right)$, where $I$ is an ideal in $C\left(X\right)$, we investigate a wide class of examples of subrings of $C\left(X\right)$ in which the sum of $z$-ideals need not be a $z$-ideal. It is observed that, for every ideal $I$ in $C\left(X\right)$, the sum of any two $z$-ideals in $I+ℝ$ is a $z$-ideal in $I+ℝ$ or all of $I+ℝ$ if and only if $X$ is an $F$-space. This result answers a question raised by Azarpanah, Namdari and Olfati in J. Commut. Algebra 11:4 (2019), 479–509.